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Convergence/Divergence of Integrals

  1. Apr 8, 2005 #1
    Determine whether the following integral converges or diverges without evaluating it.

    [tex]\int_{0}^{1-}\frac{dx}{\sqrt{1-x^4}}[/tex]

    Thanks for your help.
     
  2. jcsd
  3. Apr 8, 2005 #2

    dextercioby

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    That function behaves ~1/x^{2},so it should be integrable in that domain.U have to find a function which is greater than your function,however,it's integral on [0,1] needs to be finite...

    [tex] \int_{0}^{1} \frac{dx}{\sqrt{1-x^{4}}} =\frac{1}{4}\left(\sqrt{\pi}\right)^{3}\frac{\sqrt{2}}{\Gamma^{2}\left(\frac{3}{4}\right)} [/tex]

    Daniel.
     
  4. Apr 8, 2005 #3

    Hurkyl

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    The problem point is x=1, not x=0. √(1 - x4) behaves like 2√(1-x) there.

    P.S. 1/x2 isn't integrable over [0, 1]. :tongue2:
     
  5. Apr 8, 2005 #4

    dextercioby

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    I didn't say "integrable".And i didn't say =1/x^{2}...:wink:

    Daniel.
     
  6. Apr 8, 2005 #5

    mathwonk

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    i believe it certainly converges. now let me think why i say that.

    if y^2 = 1-x^4 then 2ydy = -4x^3 dx, so dx/sqrt(1-x^4) = dx/y = -2dy/4x^3. which makes perfectly good sense at x = 1.


    huh???

    formulas have always seemed like magic to me.
     
    Last edited: Apr 8, 2005
  7. Apr 9, 2005 #6

    dextercioby

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    Yeah,but it's still infinite,because your limits of integration will still involve 0...

    Daniel.
     
  8. Apr 9, 2005 #7

    saltydog

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    Can you explain how this integral is evaluated in terms of the Gamma function?
     
  9. Apr 9, 2005 #8

    dextercioby

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    I dunno.Basically,it returned my a value for the Legendre elliptic integral and then expressed this one in terms of Gamma factor...

    Daniel.
     
  10. Apr 9, 2005 #9

    dextercioby

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    If u make a substitution,u can transform the integral to

    [tex] \int_{1}^{+\infty} \frac{dx}{\sqrt{x^{4}-1}} [/tex]

    whose antiderivative,Mathematica finds

    Daniel.

    P.S.My Maple gave me the results...
     

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